int operatorname{cosec}^5 x , dx = | vedclass

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(C) Let $I = \int \operatorname{cosec}^5 x \, dx = \int \operatorname{cosec}^3 x \cdot \operatorname{cosec}^2 x \, dx$. Using integration by parts, le

Vedclass · Accepted Answer

$-\frac{\operatorname{cosec}^3 x \cot x}{4} - \frac{3}{8} \operatorname{cosec} x \cot x + \frac{3}{8} \log \left|	an \frac{x}{2}ight| + c$

Vedclass · Answer

(C) Let $I = \int \operatorname{cosec}^5 x \, dx = \int \operatorname{cosec}^3 x \cdot \operatorname{cosec}^2 x \, dx$. Using integration by parts, let $u = \operatorname{cosec}^3 x$ and $dv = \operatorname{cosec}^2 x \, dx$. Then $du = 3 \operatorname{cosec}^2 x (-\operatorname{cosec} x \cot x) \, dx$ and $v = -\cot x$. $I = \operatorname{cosec}^3 x(-\cot x) - \int (-\cot x) (-3 \operatorname{cosec}^3 x \cot x) \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^3 x \cot^2 x \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^3 x (\operatorname{cosec}^2 x - 1) \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3 \int \operatorname{cosec}^5 x \, dx + 3 \int \operatorname{cosec}^3 x \, dx$ $I = -\operatorname{cosec}^3 x \cot x - 3I + 3I_1$, where $I_1 = \int \operatorname{cosec}^3 x \, dx$. $4I = -\operatorname{cosec}^3 x \cot x + 3I_1$. For $I_1 = \int \operatorname{cosec}^3 x \, dx$, using integration by parts: $I_1 = \operatorname{cosec} x(-\cot x) - \int (-\cot x)(-\operatorname{cosec} x \cot x) \, dx = -\operatorname{cosec} x \cot x - \int \operatorname{cosec} x(\operatorname{cosec}^2 x - 1) \, dx$ $I_1 = -\operatorname{cosec} x \cot x - I_1 + \log|	an \frac{x}{2}| \implies 2I_1 = -\operatorname{cosec} x \cot x + \log|	an \frac{x}{2}|$. Substituting $I_1$ into the expression for $4I$: $4I = -\operatorname{cosec}^3 x \cot x + 3 \left( -\frac{1}{2} \operatorname{cosec} x \cot x + \frac{1}{2} \log|	an \frac{x}{2}| ight)$ $4I = -\operatorname{cosec}^3 x \cot x - \frac{3}{2} \operatorname{cosec} x \cot x + \frac{3}{2} \log|	an \frac{x}{2}|$ $I = -\frac{\operatorname{cosec}^3 x \cot x}{4} - \frac{3}{8} \operatorname{cosec} x \cot x + \frac{3}{8} \log|	an \frac{x}{2}| + C$.

$\int \operatorname{cosec}^5 x \, dx =$

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